Similarly, Republican Sen. Roger Wicker of Mississippi said he likes the “outlines” of a plan put forward by Sen. Susan Collins, Maine Republican, that would reopen the government and extend the debt limit for a few months while delaying the medical device tax tied to Obamacare.

“I think we’re in the ballpark there,” Mr. Wicker said.

But if Senate Majority Leader Harry Reid wants to exceed the spending caps set by the sequester deal in 2011, Mr. Wicker said, “we are indeed in for a long siege.”

Republicans say Democrats have no right to roll back spending cuts that were written into law, especially after their party was lambasted for trying to dismantle President Obama’s health care law.

But both sides agree that patience is wearing thin among their constituents.

Mr. Begich said 8,000 Alaskans are furloughed because of the government shutdown, while crabbers in his state are not on the water because they cannot obtain permits.